Logistic Regression

Journal articles we are using

Primary paper:

• Stephenson, J., Bunce, C., Doré, C. J., et. al. Ophthalmic Statistics Note 11: Logistic Regression. Br J Ophthalmol 100, 1594-1595 (2016).

Uses data from:

• Wakely, L., Rahman, R. & Stephenson, J. A comparison of several methods of macular hole measurement using optical coherence tomography, and their value in predicting anatomical and visual outcomes. Br J Ophthalmol 96, 1003-1007 (2012).

Concepts you should understand

These all play a role in logistic regression, but they are nothing more than versions of one another:

A few basic formulae

Research question

Is macular hole thickness a predictor of surgical repair success.

Regression equation for the logistic regression model was:

\[ \text{logit} \; p = 10.89 - 0.016 \; \text{x} \;\text{macular hole thickness}\;(\mu m) \]

Simulate some data

Simulate data based on this regression equation. Use:

n = 1000.

Macular hole thickness: mean = 486, sd = 142.

n <- 1000                   
set.seed(1234)
mac_hole_thickness  <-  rnorm(n, 486, 142)              # generate macular hole inner opening data with mean 486 and sd = 152
logodds_success  <-  10.89 - 0.016 * mac_hole_thickness # generate variable that is linear combination of intercept = 10.89 and coefficient for macular hole -0.016 (logit scale)
odds_success  <-  exp(logodds_success)                  # exponentiate logodds to get odds
prob_success  <-  odds_success/(1 + odds_success)       # generate probabilities from this
y_success  <-  rbinom(n, 1, prob_success)               # generate outcome variable as a function of those probabilities
df <-  data.frame(cbind(mac_hole_thickness, 
                        logodds_success, 
                        prob_success, 
                        y_success))
df <- df |> 
  filter(mac_hole_thickness > 100)                      # only include those with thickness > 100

The data

Plot the observed data

What if we fit a linear model to these data

Problem - predictions are made outside the probability domain

Putting it all together

Test